Into Multiplier Hopf ($*$-)Graph Algebras

This paper is concerned with the structures introduced recently by the authors of the current paper concerning the multiplier Hopf $*$-graph algebras and also the Cuntz-Krieger algebras and their relations with the $C^*$-graph algebras, and once again by using the $C^*$-graph algebra constructions associated to our toy example, to initiate our first class of examples concerning the multiplier Hopf $*$-graph algebras. At the final part of the paper, we apply our study to the $SL(n)$ case over the field of complex numbers, and prove that $(\mathcal{O}(SL(n),\Delta)$ possesses the initial requirements of being a discrete quantum group in the sense of Van Daele, and propose a direction in approaching one step further to the problem raised by Wang, asking ``if finite groups of Lie type have an analogue of $q$-deformations into finite quantum groups?''.